Is n = 2m + 3? (1) (n - 3)^2 = 4m^2 (2) n^2 = 4m^2 + 12m + 9

You yourself have mentioned in the solution that there are 2 values one is positive and other is negative. If i a statement is giving two possible answers, it's clearly not sufficient.

The question asks if n = 2m +3. The solution states that n can hold this value. The question never states that n should only have this value.

Statement I;

Take m = 0, n = 3.

Take m = -1, n = 1,5.... Insufficient.

Statement II:

Take m = 0, n = 3,-3... Insufficient.

Combining the two Statements,

n = 2m + 3.

You're correct to say that statement 1 yields 2 possible solutions (in red above)
However, if N = 2m + 3, then the answer to the target question is "YES, n does equal 2m + 3)
And if N = 3 - 2m , then the answer to the target question is "NO, n does not equal 2m + 3)

Since we have 2 DIFFERENT answers to the target question, the statement is NOT sufficient.

n = 2m + 3 <=> n - 3 = 2m

1) --> |n-3| = |4m| --> insufficient
2) --> |n| = |2m + 3| --> insufficient

1-2) --> (n-3)^2 - n^2 = (4m)^2 - (2m+3) ^2
<=> n = 2m + 3

--> sufficient --> C

Is \(n= 2m+ 3\) ?

1)\((n-3)^2\) = \(4m^2\)
2)\(n^2\) =\(4m^2 +12m+9\)

Bunuel , what's wrong with the following?

Is \(n= 2m+ 3\) ? => squaring both sides: is \(n^2= 4m^2+ 6m+9\) ?

Then we get a decisive answer for both statements => answer choice is D.

Whenever you raise both sides of an equation to an even power, you might get extraneous solutions. This is because raising both sides of an equation to an even power is not a reversible operation. For instance, if x = y, then x^2 = y^2. However, if x^2 = y^2, it doesn't necessarily mean x = y; it could also mean x = -y.

Take the equation x = -1 as an example. If we square both sides of the equation, we get x^2 = 1, which gives solutions x = 1 or x = -1. So, we get an extraneous solution x = 1.

Similarly, the question "Is n = 2m + 3?" cannot be directly translated to "Is n^2 = 4m^2 + 6m + 9?". This is because if from (1) n = 2m + 3, the answer is YES. However, if -n = 2m + 3, the answer is NO.

Hope it helps.

This is an awesome explanation. Thanks!